IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v66y2013icp55-69.html
   My bibliography  Save this article

Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification

Author

Listed:
  • Bachoc, François

Abstract

The Maximum Likelihood (ML) and Cross Validation (CV) methods for estimating covariance hyper-parameters are compared, in the context of Kriging with a misspecified covariance structure. A two-step approach is used. First, the case of the estimation of a single variance hyper-parameter is addressed, for which the fixed correlation function is misspecified. A predictive variance based quality criterion is introduced and a closed-form expression of this criterion is derived. It is shown that when the correlation function is misspecified, the CV does better compared to ML, while ML is optimal when the model is well-specified. In the second step, the results of the first step are extended to the case when the hyper-parameters of the correlation function are also estimated from data.

Suggested Citation

  • Bachoc, François, 2013. "Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 55-69.
    • Handle: RePEc:eee:csdana:v:66:y:2013:i:c:p:55-69
    DOI: 10.1016/j.csda.2013.03.016
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947313001187
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2013.03.016?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zhang, Hao, 2004. "Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 250-261, January.
    2. Hao Zhang & Dale L. Zimmerman, 2005. "Towards reconciling two asymptotic frameworks in spatial statistics," Biometrika, Biometrika Trust, vol. 92(4), pages 921-936, December.
    3. Paulo, Rui & García-Donato, Gonzalo & Palomo, Jesús, 2012. "Calibration of computer models with multivariate output," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3959-3974.
    4. Andrianakis, Ioannis & Challenor, Peter G., 2012. "The effect of the nugget on Gaussian process emulators of computer models," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4215-4228.
    5. Ying, Zhiliang, 1991. "Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 280-296, February.
    6. Marrel, Amandine & Iooss, Bertrand & Van Dorpe, François & Volkova, Elena, 2008. "An efficient methodology for modeling complex computer codes with Gaussian processes," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4731-4744, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    2. Marrel, Amandine & Iooss, Bertrand, 2024. "Probabilistic surrogate modeling by Gaussian process: A review on recent insights in estimation and validation," Reliability Engineering and System Safety, Elsevier, vol. 247(C).
    3. Betancourt, José & Bachoc, François & Klein, Thierry & Idier, Déborah & Pedreros, Rodrigo & Rohmer, Jérémy, 2020. "Gaussian process metamodeling of functional-input code for coastal flood hazard assessment," Reliability Engineering and System Safety, Elsevier, vol. 198(C).
    4. Kleijnen, Jack P.C. & Mehdad, E., 2013. "Conditional simulation for efficient global optimization," Other publications TiSEM 52e4860d-9887-4a63-b19a-7, Tilburg University, School of Economics and Management.
    5. Lee, Dongjin & Kramer, Boris, 2023. "Multifidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos-Kriging," Reliability Engineering and System Safety, Elsevier, vol. 235(C).
    6. Gerber, Florian & Nychka, Douglas W., 2021. "Parallel cross-validation: A scalable fitting method for Gaussian process models," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    7. Aikaterini P. Kyprioti & Alexandros A. Taflanidis & Norberto C. Nadal-Caraballo & Madison O. Campbell, 2021. "Incorporation of sea level rise in storm surge surrogate modeling," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 105(1), pages 531-563, January.
    8. Binois, M. & Ginsbourger, D. & Roustant, O., 2015. "Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations," European Journal of Operational Research, Elsevier, vol. 243(2), pages 386-394.
    9. Bachoc, François & Lagnoux, Agnès & Nguyen, Thi Mong Ngoc, 2017. "Cross-validation estimation of covariance parameters under fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 42-67.
    10. Marrel, Amandine & Iooss, Bertrand, 2024. "Probabilistic surrogate modeling by Gaussian process: A new estimation algorithm for more robust prediction," Reliability Engineering and System Safety, Elsevier, vol. 247(C).
    11. Cousin, Areski & Maatouk, Hassan & Rullière, Didier, 2016. "Kriging of financial term-structures," European Journal of Operational Research, Elsevier, vol. 255(2), pages 631-648.
    12. Acharki, Naoufal & Bertoncello, Antoine & Garnier, Josselin, 2023. "Robust prediction interval estimation for Gaussian processes by cross-validation method," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    13. Jize Zhang & Alexandros A. Taflanidis & Norberto C. Nadal-Caraballo & Jeffrey A. Melby & Fatimata Diop, 2018. "Advances in surrogate modeling for storm surge prediction: storm selection and addressing characteristics related to climate change," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 94(3), pages 1225-1253, December.
    14. Bachoc, François & Bevilacqua, Moreno & Velandia, Daira, 2019. "Composite likelihood estimation for a Gaussian process under fixed domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    15. Auffray, Yves & Barbillon, Pierre & Marin, Jean-Michel, 2014. "Bounding rare event probabilities in computer experiments," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 153-166.
    16. Bachoc, François, 2014. "Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 1-35.
    17. Cousin, Areski & Maatouk, Hassan & Rullière, Didier, 2016. "Kriging of financial term-structures," European Journal of Operational Research, Elsevier, vol. 255(2), pages 631-648.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bachoc, François & Lagnoux, Agnès & Nguyen, Thi Mong Ngoc, 2017. "Cross-validation estimation of covariance parameters under fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 42-67.
    2. Wenpin Tang & Lu Zhang & Sudipto Banerjee, 2021. "On identifiability and consistency of the nugget in Gaussian spatial process models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(5), pages 1044-1070, November.
    3. Bachoc, François, 2014. "Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 1-35.
    4. Wu, Wei-Ying & Lim, Chae Young & Xiao, Yimin, 2013. "Tail estimation of the spectral density for a stationary Gaussian random field," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 74-91.
    5. Zhang, Tonglin, 2017. "An example of inconsistent MLE of spatial covariance parameters under increasing domain asymptotics," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 108-113.
    6. Werner Müller & Milan Stehlík, 2009. "Issues in the optimal design of computer simulation experiments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(2), pages 163-177, March.
    7. Bevilacqua, Moreno & Caamaño-Carrillo, Christian & Porcu, Emilio, 2022. "Unifying compactly supported and Matérn covariance functions in spatial statistics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    8. Arthur P. Guillaumin & Adam M. Sykulski & Sofia C. Olhede & Frederik J. Simons, 2022. "The Debiased Spatial Whittle likelihood," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1526-1557, September.
    9. Ganggang Xu & Marc G. Genton, 2017. "Tukey -and- Random Fields," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1236-1249, July.
    10. Chevalier, Clément & Picheny, Victor & Ginsbourger, David, 2014. "KrigInv: An efficient and user-friendly implementation of batch-sequential inversion strategies based on kriging," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1021-1034.
    11. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    12. Christopher J. Geoga & Mihai Anitescu & Michael L. Stein, 2021. "Flexible nonstationary spatiotemporal modeling of high‐frequency monitoring data," Environmetrics, John Wiley & Sons, Ltd., vol. 32(5), August.
    13. Keshavarz, Hossein & Scott, Clayton & Nguyen, XuanLong, 2016. "On the consistency of inversion-free parameter estimation for Gaussian random fields," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 245-266.
    14. Maitreyee Bose & James S. Hodges & Sudipto Banerjee, 2018. "Toward a diagnostic toolkit for linear models with Gaussian‐process distributed random effects," Biometrics, The International Biometric Society, vol. 74(3), pages 863-873, September.
    15. Lim, Chae Young & Chen, Chien-Hung & Wu, Wei-Ying, 2017. "Numerical instability of calculating inverse of spatial covariance matrices," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 182-188.
    16. Bachoc, François & Bevilacqua, Moreno & Velandia, Daira, 2019. "Composite likelihood estimation for a Gaussian process under fixed domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    17. Rajala, T. & Penttinen, A., 2014. "Bayesian analysis of a Gibbs hard-core point pattern model with varying repulsion range," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 530-541.
    18. Girard, Didier A., 2016. "Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models — I: Densely sampled processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 191-197.
    19. Lu, Zudi & Tjostheim, Dag & Yao, Qiwei, 2008. "Spatial smoothing, Nugget effect and infill asymptotics," LSE Research Online Documents on Economics 24133, London School of Economics and Political Science, LSE Library.
    20. Veiga, Sébastien Da & Marrel, Amandine, 2020. "Gaussian process regression with linear inequality constraints," Reliability Engineering and System Safety, Elsevier, vol. 195(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:66:y:2013:i:c:p:55-69. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.